Creating a path with differential equations
When I have understood you correctly, you have already the solution of your diff eq, and you know x0, y0, z0 as parameters of this solution, defining the start location. When you run under Python a loop by incrementing t in discrete steps, you will be able to create a 3d chain of vertices with these coordinates and use this polygon as path for animation (convert to a curve, if needed). Since I am a newbie myself, I cannot provide any code, however.

Greetings from Germany, Tschöbbel